Insurance with a deductible: a way out of the long term care insurance puzzle

Abstract

Long-term care (LTC) is one of the largest uninsured risks facing the elderly. In this paper, we first survey the standard causes of what has been dubbed the LTC insurance puzzle and then suggest that a possible way out of this puzzle is to make the reimbursement formula less threatening for those who fear a too long period of dependence. We adopt a reimbursement formula resting on Arrow’s theorem of the deductible, i.e. that it is optimal to focus insurance coverage on the states with largest expenditures. It implies full self-insurance for the first years of dependency followed by full insurance thereafter. We show that this result remains at work with ex post moral hazard.

Appendix: Model with several periods

In this appendix, we show why the states of nature used in the above model can be viewed as periods of dependency. We do so using a simple example. Assume 3 periods consisting of the three sequences:

• AAA, with probability $p_{0}=(1-p)^{2}$

• AAD, with $p_{1}=(1-p)p$

• ADD with $p_{2}=p$

where A stands for autonomy and D for dependence. We assume that the probability of dependence p is constant and that the state of dependance is irreversible. We want to show that the above model of one period and different states of nature is a reduced form of a multiple period model. We assume no time preference and a zero rate of interest and constant income per period equal to y. People can freely save (s in the first period and $\sigma$ in the second period).

We can now write the lifetime utilities corresponding to these 3 sequences.

$$\begin{aligned} U_{0}=& {} u(w-\pi -s_{0})+u(y-\pi +s_{0}-\sigma _{0}) +u\left( y-\pi +\sigma _{0}\right) +3H(A) \\ U_{1}=& {} u(w-\pi -s_{1})+u(y-\pi +s_{1}-\sigma _{1}) +u\left( y-\pi -\left( 1-\alpha _{1}\right) D_{1}+\sigma _{1}\right) \\&+2H(A)+H(A-L+D_{1}) \\ U_{2}=& {} u(w-\pi -s_{2})+u(y-\pi +s_{2}-\sigma _{2} -\left( 1-\alpha _{2}\right) D_{2}) \\&\quad +u\left( y-\pi -\left( 1-\alpha _{2}\right) D_{2} +\sigma _{2}\right) +H(A)+2H(A-L+D_{2}) \end{aligned}$$